# How to Choose the Refueling of New Energy Vehicles under Swapping vs. Charging Mode: From the Consumers’ Perspective

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Literature Review

#### 2.1. Research on the Battery Swapping/Charging Mode for NEVs

#### 2.2. Research on the Extended Warranty Service

## 3. Theoretical Models

#### 3.1. Base Model

**Proposition**

**1.**

**Lemma**

**1.**

**Theorem**

**1.**

**Corollary**

**1.**

#### 3.2. Model with Extended Warranty Service

**Proposition**

**2.**

**Theorem**

**2.**

- (i)
- When $0<\frac{\delta}{k}<(2\alpha /\beta {)}^{\genfrac{}{}{0pt}{}{1}{2}},{\lambda >k}^{2}/\beta $ hold on, the NEV-SM maker’s profit is more advantageous that the NEV-CM maker’s, whereas the NEV-SM maker possesses less market share than its rival. (i.e., $\stackrel{~}{{D}_{1}^{*}}<\stackrel{~}{{D}_{2}^{*}},\stackrel{~}{{{\pi}_{1}}^{*}}>\stackrel{~}{{{\pi}_{2}}^{*}}$);
- (ii)
- When $\frac{\delta}{k}>(2\alpha /\beta {)}^{\genfrac{}{}{0pt}{}{1}{2}},\lambda \left(\beta {\delta}^{2}-\alpha {k}^{2}\right)/2\alpha \beta $ hold on, the NEV-CM maker is more competitive than the NEV-SM maker (i.e., $\stackrel{~}{{D}_{1}^{*}}<\stackrel{~}{{D}_{2}^{*}},\stackrel{~}{{{\pi}_{1}}^{*}}\stackrel{~}{{{\pi}_{2}}^{*}}$).

**Corollary**

**2.**

- (i)
- When $2\alpha {k}^{2}>\beta {\delta}^{2},{\lambda k}^{2}/\beta $ , both $\stackrel{~}{m}$ and $T$ increase with $\delta $;
- (ii)
- When $2\alpha {k}^{2}<\beta {\delta}^{2},\lambda \left(\beta {\delta}^{2}-\alpha {k}^{2}\right)/2\alpha \beta $, $\stackrel{~}{m}$ increases with $\delta $; while $T$ decreases with $\delta $.

## 4. The Impact of the Subsidy Policy

**Lemma**

**2.**

**Lemma**

**3.**

- (i)
- When $\alpha {k}^{2}>\beta {\delta}^{2}+2a\beta \delta ,\lambda \left(\alpha {k}^{2}-a\beta \delta \right)/\alpha \beta $ or $\alpha {k}^{2}<\beta {\delta}^{2},\lambda \left(\beta {\delta}^{2}-\alpha {k}^{2}+a\beta \delta \right)/\alpha \beta $ is satisfied, for the NEV-SM maker, the optimal price and demand at the preliminary stage I is higher than at the mature stage II (i.e.,${\stackrel{~}{{p}_{1}}}^{\mathrm{\Pi}}{\stackrel{~}{{p}_{1}}}^{\mathrm{{\rm I}}}$, ${\stackrel{~}{{D}_{1}}}^{\mathrm{\Pi}}<{\stackrel{~}{{D}_{1}}}^{\mathrm{{\rm I}}}$);
- (ii)
- When $\lambda \alpha +b\delta >\alpha {k}^{2}/\beta >{\delta}^{2}+2b\delta $ or ${\delta}^{2}>\alpha {k}^{2}/\beta >{\delta}^{2}-\alpha \lambda +b\delta $ hold on, for the NEV-SM maker, the optimal price and demand at the mature stage II is larger than at the preliminary stage I (i.e., ${\stackrel{~}{{p}_{2}}}^{\mathrm{\Pi}}>{\stackrel{~}{{p}_{2}}}^{\mathrm{{\rm I}}},{\stackrel{~}{{D}_{2}}}^{\mathrm{\Pi}}{\stackrel{~}{{D}_{2}}}^{\mathrm{{\rm I}}}$).

**Corollary**

**3.**

- (i)
- When $\lambda >\left({\delta}^{2}+2a\delta \right)/\alpha $ (for stage II, $\lambda >\left({\delta}^{2}+2b\delta \right)/\alpha $) is satisfied, the NEV-SM maker’s optimal price and demand in the presence of a subsidy at stage I (II) are higher than those in the absence of a subsidy, i.e., ${p}_{1}^{\mathrm{{\rm I}}}>{p}_{1},{D}_{1}^{\mathrm{{\rm I}}}{D}_{1}$ (${p}_{1}^{\mathrm{\Pi}}>{p}_{1},{D}_{1}^{\mathrm{\Pi}}{D}_{1}$); while for the NEV-CM maker, the optimal price and demand in the presence of a subsidy at stage I (II) are lower than those in the absence of a subsidy, i.e., ${p}_{2}^{\mathrm{{\rm I}}}<{p}_{2},{D}_{2}^{\mathrm{{\rm I}}}{D}_{2}$ (${p}_{2}^{\mathrm{\Pi}}<{p}_{2},{D}_{2}^{\mathrm{\Pi}}{D}_{2}$).
- (ii)
- When ${\lambda >k}^{2}/\beta >{(\delta}^{2}+2a\delta )/\alpha $ or $\frac{\beta {\delta}^{2}}{2}>\alpha {k}^{2}>\beta {\delta}^{2}-\lambda \alpha \beta +2a\beta \delta $ (for stage II, $\alpha {k}^{2}>\beta {\delta}^{2}+2b\beta \delta ,\lambda {k}^{2}/2\beta $ or $\alpha {k}^{2}<\beta {\delta}^{2},\lambda \left(\beta {\delta}^{2}-\alpha {k}^{2}+2b\beta \delta \right)/\alpha \beta $) holds on, the NEV-SM maker’s optimal price and demand in the presence of a subsidy at stage I (II) are lower than those in the absence of a subsidy, i.e., ${\stackrel{~}{{p}_{1}}}^{\mathrm{\Pi}}<\stackrel{~}{{p}_{1},}{\stackrel{~}{{D}_{1}}}^{\mathrm{{\rm I}}}\stackrel{~}{{D}_{1}}$ (${\stackrel{~}{{p}_{1}}}^{\mathrm{\Pi}}<\stackrel{~}{{p}_{1},}{\stackrel{~}{{D}_{1}}}^{\mathrm{\Pi}}\stackrel{~}{{D}_{1}}$); while for the NEV-CM maker, the optimal price, the duration of the extended warranty service, and demand in the presence of a subsidy at stage I (II) are higher than those in the absence of a subsidy, i.e., ${\stackrel{~}{{p}_{2}}}^{\mathrm{{\rm I}}}>\stackrel{~}{{p}_{2},}{\stackrel{~}{{D}_{2}}}^{\mathrm{{\rm I}}}\stackrel{~}{{D}_{2}},{\stackrel{~}{T}}^{\mathrm{{\rm I}}}\stackrel{~}{T}$ (${\stackrel{~}{{p}_{2}}}^{\mathrm{\Pi}}>\stackrel{~}{{p}_{2},}{\stackrel{~}{{p}_{2}}}^{\mathrm{\Pi}}\stackrel{~}{{D}_{2}},{\stackrel{~}{T}}^{\mathrm{\Pi}}\stackrel{~}{T}$).

## 5. Numerical Study

#### 5.1. The Impact on Consumer Surplus and Social Welfare

#### 5.2. The Impact on Performance

- (1)
- The single effect of α and$\delta $on profits

- (2)
- The single effect of β and k on profits

_{1}, below which the lowest profit occurs to the NEV-CM maker in the case where the swapping network has taken shape; otherwise, it happens when the swapping network is at an infant stage. In contrast, there is another certain value k

_{2}, above which the highest profit occurs to the NEV-SM maker in the case where the swapping network has taken shape; otherwise, it happens when the swapping network is at an initial stage.

- (3)
- The joint effect on profits

## 6. Conclusions

- (1)
- In the absence of an extended warranty service, there exists an optimal equilibrium condition for both carmakers, in which the NEV-SM maker adopts the low-price strategy at the initial development stage of NEV-SM, with aims to tap into the market and attract potential consumers, yet this move is not a long-term solution due to the fact that the NEV-CM maker invariably occupies more market share and profit in the NEV market.
- (2)
- In the presence of an extended warranty service, the NEV-CM maker does not necessarily dominate the market. Consumers’ choice of two kinds of products relies on two critical factors, namely $\lambda \mathrm{a}\mathrm{n}\mathrm{d}\frac{\delta}{k}$. When $0<\frac{\delta}{k}<(2\alpha /\beta {)}^{\genfrac{}{}{0pt}{}{1}{2}},{\lambda >k}^{2}/\beta $, the NEV-SM maker’s profit does surpass the NEV-CM maker’s. Nevertheless, the extended warranty service ultimately enhances the NEV-CM maker’s dominance.
- (3)
- In the absence of an extended warranty service, when the subsidy standards a and b are below a specific threshold value (i.e., $\left(\lambda \alpha -{\delta}^{2}\right)/2\delta $), both the demand and price for the NEV-SM maker at stage I are larger than those at stage II, whereas for the NEV-CM maker, the outcome is the opposite, which shows that government subsidies have little effect on NEV-SM manufacturers at stage II if the government subsidy’s size reduces to a degree.
- (4)
- In the presence of an extended warranty service, the government subsidy can dramatically increase the NEV-SM maker’s market share at stage I. Meanwhile, the numerical studies show that the NEV-SM maker’s profit, consumer surplus, and social welfare have been improved by implementing the subsidy policy, which implies that the subsidy is a critical factor in propelling the diffusion of the swapping mode at the initial stage.

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A

Notations | Definition |
---|---|

Decisions | |

M | The number of battery-swapping stations |

T | The duration of the extended warranty service |

${p}_{i}$ | Product price of NEVs with i mode (i = 1 refers to SM, i = 2 stands for CM) |

Parameters | |

$\alpha $ | The cost coefficient of battery-swapping stations |

$\beta $ | The cost coefficient of the extended warranty service |

$\delta $ | The battery swapping station’s convenience coefficient |

k | The extended warranty service coefficient |

$\lambda $ | The valuation incremental rate of CM relative to SM |

s | The amount of the government subsidy |

${u}_{i}$ | Consumer utility of NEVs with i mode (i = 1 refers to SM, i = 2 stands for CM) |

$v$ | Consumer valuation of the NEV-SM product |

${c}_{i}$ | Unit product cost of NEVs with i mode (i = 1 refers to SM, i = 2 stands for CM) |

${D}_{i}$ | Consumer demand of NEVs with i mode (i = 1 refers to SM, i = 2 stands for CM) |

${\pi}_{i}$ | Profit of NEVs with i mode (i = 1 refers to SM, i = 2 stands for CM) |

Without the Extended Warranty Service | With the Extended Warranty Service | |||||
---|---|---|---|---|---|---|

Stage I | Stage II | Stage I | Stage II | |||

Extended warranty service duration ${{T}^{*}}^{\mathrm{{\rm I}},\mathrm{\Pi}}$ | $-$ | $-$ | ${{\stackrel{~}{T}}^{*}}^{\mathrm{{\rm I}},\mathrm{\Pi}}$ | $\frac{-k\left(2\alpha \lambda -{\delta}^{2}-a\delta \right)}{{\alpha k}^{2}+\beta {\delta}^{2}-3\alpha \beta \lambda}$ | $\frac{-k\left(2\alpha \lambda -{\delta}^{2}-b\delta \right)}{{\alpha k}^{2}+\beta {\delta}^{2}-3\alpha \beta \lambda}$ | |

Number of swapping stations ${{m}^{*}}^{\mathrm{{\rm I}},\mathrm{\Pi}}$ | $\frac{\lambda \left(3a+\delta \right)}{3\alpha \lambda -{\delta}^{2}}$ | $\frac{\lambda \left(3b+\delta \right)}{3\alpha \lambda -{\delta}^{2}}$ | ${{\stackrel{~}{m}}^{*}}^{\mathrm{{\rm I}},\mathrm{\Pi}}$ | $\frac{{k}^{2}\left(a+\delta \right)-\beta \lambda \left(3a+\delta \right)}{{\alpha k}^{2}+\beta {\delta}^{2}-3\alpha \beta \lambda}$ | $\frac{{k}^{2}\left(b+\delta \right)-\beta \lambda \left(3b+\delta \right)}{{\alpha k}^{2}+\beta {\delta}^{2}-3\alpha \beta \lambda}$ | |

Price | ${{{p}_{1}}^{*}}^{\mathrm{{\rm I}},\mathrm{\Pi}}$ | $\frac{\lambda \left(a\delta +\alpha \lambda \right)+c\left(3\alpha \lambda -{\delta}^{2}\right)}{3\alpha \lambda -{\delta}^{2}}$ | $\frac{\lambda \left(b\delta +\alpha \lambda \right)+c\left(3\alpha \lambda -{\delta}^{2}\right)}{3\alpha \lambda -{\delta}^{2}}$ | ${{\stackrel{~}{{p}_{1}}}^{*}}^{\mathrm{{\rm I}},\mathrm{\Pi}}$ | $\left[-\alpha \beta \lambda \left(3c+\lambda \right)+\alpha {k}^{2}\left(c+\lambda \right)-\beta \delta \left(a\lambda -c\delta \right)\right]/\left({\alpha k}^{2}+\beta {\delta}^{2}-3\alpha \beta \lambda \right)$ | $\left[-\alpha \beta \lambda \left(3c+\lambda \right)+\alpha {k}^{2}\left(c+\lambda \right)-\beta \delta \left(b\lambda -c\delta \right)\right]/\left({\alpha k}^{2}+\beta {\delta}^{2}-3\alpha \beta \lambda \right)$ |

${{{p}_{2}}^{*}}^{\mathrm{{\rm I}},\mathrm{\Pi}}$ | $\frac{\lambda \left(2\alpha \lambda -{\delta}^{2}-a\delta \right)+c\left(3\alpha \lambda -{\delta}^{2}\right)}{3\alpha \lambda -{\delta}^{2}}$ | $\frac{\lambda \left(2\alpha \lambda -{\delta}^{2}-b\delta \right)+c\left(3\alpha \lambda -{\delta}^{2}\right)}{3\alpha \lambda -{\delta}^{2}}$ | ${{\stackrel{~}{{p}_{2}}}^{*}}^{\mathrm{{\rm I}},\mathrm{\Pi}}$ | $\left[-2\alpha \beta {\lambda}^{2}+c\left(\alpha {k}^{2}+\beta {\delta}^{2}\right)+\lambda \beta \left(-3c\alpha +a\delta +{\delta}^{2}\right)\right]/\left({\alpha k}^{2}+\beta {\delta}^{2}-3\alpha \beta \lambda \right)$ | $\left[-2\alpha \beta {\lambda}^{2}+c\left(\alpha {k}^{2}+\beta {\delta}^{2}\right)+\lambda \beta \left(-3c\alpha +b\delta +{\delta}^{2}\right)\right]/\left({\alpha k}^{2}+\beta {\delta}^{2}-3\alpha \beta \lambda \right)$ | |

Demand | ${{D}_{1}^{*}}^{\mathrm{{\rm I}},\mathrm{\Pi}}$ | $\frac{a\delta +\alpha \lambda}{3\alpha \lambda -{\delta}^{2}}$ | $\frac{b\delta +\alpha \lambda}{3\alpha \lambda -{\delta}^{2}}$ | ${{\stackrel{~}{{D}_{1}}}^{*}}^{\mathrm{{\rm I}},\mathrm{\Pi}}$ | $\frac{-\left(a\beta \delta +\alpha \beta \lambda -\alpha {k}^{2}\right)}{{\alpha k}^{2}+\beta {\delta}^{2}-3\alpha \beta \lambda}$ | $\frac{-\left(b\beta \delta +\alpha \beta \lambda -\alpha {k}^{2}\right)}{{\alpha k}^{2}+\beta {\delta}^{2}-3\alpha \beta \lambda}$ |

${{D}_{2}^{*}}^{\mathrm{{\rm I}},\mathrm{\Pi}}$ | $\frac{2\alpha \lambda -{\delta}^{2}-a\delta}{3\alpha \lambda -{\delta}^{2}}$ | $\frac{2\alpha \lambda -{\delta}^{2}-b\delta}{3\alpha \lambda -{\delta}^{2}}$ | ${{\stackrel{~}{{D}_{2}}}^{*}}^{\mathrm{{\rm I}},\mathrm{\Pi}}$ | $\frac{-\beta \left(2\alpha \lambda -{\delta}^{2}-a\delta \right)}{{\alpha k}^{2}+\beta {\delta}^{2}-3\alpha \beta \lambda}$ | $\frac{-\beta \left(2\alpha \lambda -{\delta}^{2}-b\delta \right)}{{\alpha k}^{2}+\beta {\delta}^{2}-3\alpha \beta \lambda}$ | |

Profit | ${{{\pi}_{1}}^{*}}^{\mathrm{{\rm I}},\mathrm{\Pi}}$ | $\lambda \left[2{\left(\alpha \lambda \right)}^{2}+\alpha \lambda \left(9{a}^{2}+4a\delta -{\delta}^{2}\right)-2a{\delta}^{2}\left(2a+\delta \right)\right]/\left[2{\left(3\alpha \lambda -{\delta}^{2}\right)}^{2}\right]$ | $\lambda \left[2{\left(\alpha \lambda \right)}^{2}+\alpha \lambda \left(9{a}^{2}+6a\delta -{\delta}^{2}-2b\delta +18{a}^{2}-18ab\right)+6ab{\delta}^{2}-2{\delta}^{2}\left(3{a}^{2}+a\delta +2{b}^{2}\right)\right]/\left[2{\left(3\alpha \lambda -{\delta}^{2}\right)}^{2}\right]$ | ${{\stackrel{~}{{\pi}_{1}}}^{*}}^{\mathrm{{\rm I}},\mathrm{\Pi}}$ | $\left\{\lambda {\beta}^{2}\left[2{\left(\alpha \lambda \right)}^{2}+\alpha \lambda \left(9{a}^{2}+4a\delta -{\delta}^{2}\right)-2a{\delta}^{2}\left(2a+\delta \right)\right]-2{\beta k}^{2}\left[2{\left(\alpha \lambda \right)}^{2}+\alpha \lambda \left(a+\delta \right)\left(3a-\delta \right)-a{\delta}^{2}\left(a+\delta \right){+\alpha k}^{4}\left({a}^{2}+2\alpha \lambda -{\delta}^{2}\right)\right]\right\}/\left[2{\left({\alpha k}^{2}+\beta {\delta}^{2}-3\alpha \beta \lambda \right)}^{2}\right]$ | $\left\{{\alpha \lambda}^{2}{\beta}^{2}\left[-{\delta}^{2}+2\delta \left(3a-b\right)\right]+{\alpha \lambda}^{2}{\beta}^{2}9{b}^{2}+6\alpha \beta \lambda {k}^{2}{b}^{2}+2\alpha \beta \lambda {k}^{2}\left[{\delta}^{2}-2\delta \left(2a-b\right)\right]-{\alpha k}^{4}\left[{\delta}^{2}-2\delta \left(a-b\right)\right]-2\beta {\delta}^{2}\left[\beta \lambda \left(a\delta +2{b}^{2}\right)\right]+{k}^{2}\left(a\delta +{b}^{2}\right)+2\lambda {\alpha}^{2}{\left({k}^{2}-\beta \lambda \right)}^{2}+{\alpha k}^{4}{b}^{2}\right\}/\left[2{\left({\alpha k}^{2}+\beta {\delta}^{2}-3\alpha \beta \lambda \right)}^{2}\right]$ |

${{{\pi}_{2}}^{*}}^{\mathrm{{\rm I}},\mathrm{\Pi}}$ | $\frac{{\lambda \left(2\alpha \lambda -{\delta}^{2}-a\delta \right)}^{2}}{{\left(3\alpha \lambda -{\delta}^{2}\right)}^{2}}$ | $\frac{{\lambda \left(2\alpha \lambda -{\delta}^{2}-b\delta \right)}^{2}}{{\left(3\alpha \lambda -{\delta}^{2}\right)}^{2}}$ | ${{\stackrel{~}{{\pi}_{2}}}^{*}}^{\mathrm{{\rm I}},\mathrm{\Pi}}$ | $\left[\beta \left(2\beta \lambda -{k}^{2}\right){\left(2\alpha \lambda -{\delta}^{2}-a\delta \right)}^{2}\right]/\left[2{\left({\alpha k}^{2}+\beta {\delta}^{2}-3\alpha \beta \lambda \right)}^{2}\right]$ | $\left[\beta \left(2\beta \lambda -{k}^{2}\right){\left(2\alpha \lambda -{\delta}^{2}-b\delta \right)}^{2}\right]/\left[2{\left({\alpha k}^{2}+\beta {\delta}^{2}-3\alpha \beta \lambda \right)}^{2}\right]$ |

c | v | k | A | b | δ | α | β | λ |
---|---|---|---|---|---|---|---|---|

1 | 0.5 | 0.2–0.6 | 0.4 | 0.1 | 0.5–0.9 | 0.6–0.8 | 0.3–0.5 | 0.7 |

**Proof**

**of**

**Proposition**

**1.**

**Proof**

**of**

**Lemma**

**1.**

**Proof**

**of**

**Theorem 1.**

**Proof**

**of**

**Corollary 1.**

**Proof**

**of**

**Proposition 2.**

**Proof**

**of**

**Theorem 2.**

**Proof**

**of**

**Corollary**

**2.**

**Proof**

**of**

**Lemma**

**2.**

**Proof**

**of**

**Lemma**

**3.**

**Proof**

**of**

**Corollary**

**3.**

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## Share and Cite

**MDPI and ACS Style**

Li, J.; Wu, D.; Wang, Y.
How to Choose the Refueling of New Energy Vehicles under Swapping vs. Charging Mode: From the Consumers’ Perspective. *World Electr. Veh. J.* **2023**, *14*, 211.
https://doi.org/10.3390/wevj14080211

**AMA Style**

Li J, Wu D, Wang Y.
How to Choose the Refueling of New Energy Vehicles under Swapping vs. Charging Mode: From the Consumers’ Perspective. *World Electric Vehicle Journal*. 2023; 14(8):211.
https://doi.org/10.3390/wevj14080211

**Chicago/Turabian Style**

Li, Jizi, Doudou Wu, and Yong Wang.
2023. "How to Choose the Refueling of New Energy Vehicles under Swapping vs. Charging Mode: From the Consumers’ Perspective" *World Electric Vehicle Journal* 14, no. 8: 211.
https://doi.org/10.3390/wevj14080211